You can get all of the elixers via gambling. In theory, it's quicker to get it via gambling because every gambling attempt will give you something, whereas you're not guaranteed to get anything at all on some missions.

Classes

Bards have a better chance of finding loot from missions

Alchemists have better chances of getting elixers via gambling

Princes/princess gets about 50-100% more gold for gambling

Kings/queens get 100-150% more gold for gambling

Difficulty

The higher the difficulty the mission, the better chances of loot in general; the higher difficulty the encounter, the better chances for loot in general. For example, legendary quests have a +6% chance of loot per hostile encounter and +3% per non-hostile encounters; a +4 (very difficult) encounter is +4% more. So in theory, very difficuly legendary encounters give you +10% chance of loot.

Loot Odds

What you GET from loot is completely random; 1d4, 1=weapon, 2=armor, 3=accessory, 4=potion. There's a 1 in 100 chance on ANY of those of getting something 'special'. If you get a special potion there's a 25% chance that it's an elixer (alchemists double that to 50%).

So, on the average legendary quest (20 ticks), with common encounters you have a 6-7% chance per encounter of getting SOMETHING; which means a (6*20) 120% chance of loot per legendary quest. Which is roughly a 1.2 to 1.4% chance per legendary quest of getting something special. In effect, every 90-100 legendary quests you should stumble across a relic/artifact/magical/elixer. So by the time you've done 500-750 legendary quests, by sheer mathematical odds, you should have found an elixer, and a relic weapon, armor, and accessory.

Granted, this is LSL odds we're talking about. You could get one your first encounter. Or never. SL isn't always nice. All of that assumes you're getting a constant supply of encounters though. There's an equally likely chance you just find some stones in the riverbed, flowers in the field, or sell off some baubles.

Relics

Current Relics (always spawns with maxed possible values for the butterfly's level). In the future, all of the below relics (and the ones added in later) will be able to be traded in for something special. We haven't decided just what that'll be, but, yeah…